Solve the equation \(64^{x+\frac{1}{3}}+2^{3 x}-3=0\).

Find the values of \(k\) for which the equation \(4 x^{2}-k=4 k x-2\) has no real roots.

(a) Solve the equation \(8^{\frac{1}{x}}-2 \times 8^{-\frac{1}{x}}=1\). (b) It is given that \((a-\sqrt{3})^{2}=b+(3-b) \sqrt{3}\), where \(a\) and \(b\) are integers. Find the possible values of \(a\) and \(b\).

(a) Solve the following simultaneous equations. \(\begin{aligned} \mathrm{e}^{x+y} \times \mathrm{e}^{3 x-2 y} & =1 \\ x^{2} y & =256 \end{aligned}\) (b) Solve the equation \(10 \mathrm{e}^{(2 x-1)}-11=6 \mathrm{e}^{(1-2 x)}\), giving your answer in exact form.

Solve the equation \(3\left(2^{2 x+1}\right)-11\left(2^{x}\right)+3=0\), giving your answers correct to 2 decimal places.

(a) Solve the equation

\(5^{2y-1}=6\times3^y,\)

giving your answer correct to 3 decimal places.

(b) Solve the equation

\(e^{2x}-4+3e^{-2x}=0,\)

giving your answers in exact form.

(a) Find the exact solution of the equation

\(2\mathrm e^{6x}-3\mathrm e^{3x}-5=0.\)

(b) Solve the simultaneous equations

\(\mathrm e^{4x-7}\div \mathrm e^{5x+7y}=\frac{1}{\mathrm e^2},\)

\(xy+18=0.\)

(a) Solve the equation \(\displaystyle \frac{625^{(x^3-1)/2}}{125^{x^3}}=5\).

(b) On the axes, sketch the graph of \(y=4\mathrm{e}^x+3\), showing the values of any intercepts with the coordinate axes.

Solve the following equations, giving your answers to 3 significant figures.

(a)

\(2^{3x+1}=5^{x-2}\)

(b)

\(e^{2y+1}=1+\frac6{e^{2y+1}}\)

Solve the equation

\(4e^{2x-3}=7e^{5-x}.\)

Write

\(\frac{(pqr)^{-2}r^{\frac13}}{(p^2r)^{-1}q^3}\)

in the form \(p^a q^b r^c\), where \(a\), \(b\) and \(c\) are constants.

Given that

\(y=2(7^{2x})-3(7^{x+1})+19,\)

find the value of \(x\) when \(y=30\).

(a) Given that

\(\frac{\sqrt[3]{xy}(zy)^2}{(xz)^{-3}\sqrt z}=x^ay^bz^c,\)

find the exact values of the constants \(a\), \(b\) and \(c\).

(b) Solve the equation

\(5\left(2^{2p+1}\right)-17\left(2^p\right)+3=0.\)

(a) Solve the simultaneous equations

\(e^x+e^y=5,\qquad 2e^x-3e^y=8.\)

(b) Solve the equation

\(e^{2t-1}=5e^{5t-3}.\)

Solve the equation

\(3^{2x}-3^{x+1}-4=0,\)

giving your answer in an exact form.

(a) Given that

\(\log_2x+2\log_4y=8,\)

find the value of \(xy\).

(b) Using the substitution \(y=2^x\), or otherwise, solve

\(2^{2x+1}-2^{x+1}-2^x+1=0.\)

(a) Solve the simultaneous equations

\(10^{x+2y}=5,\qquad 10^{3x+4y}=50,\)

giving \(x\) and \(y\) in exact simplified form.

(b) Solve

\(2x^{\frac23}-x^{\frac13}-10=0.\)

Find the value of \(x\) such that

\(\frac{4^{x+1}}{2^{x-1}}=32^{x/3}\times8^{1/3}.\)

Solve the simultaneous equations

\(3^x\times 9^{y-1}=243,\)

\(8\times 2^{y-\frac12}=\frac{2^{2x+1}}{4\sqrt2}.\)

The number, \(B\), of a certain type of bacteria at time \(t\) days is described by \(B=200e^{2t}+800e^{-2t}\).

(i) Find \(B\) when \(t=0\).

(ii) At the instant when \(\frac{dB}{dt}=1200\), show that \(e^{4t}-3e^{2t}-4=0\).

(iii) Using the substitution \(u=e^{2t}\), or otherwise, solve \(e^{4t}-3e^{2t}-4=0\).

Given that \(\dfrac{\sqrt p(qr)^{-2}}{p^2q^{1/3}r}=\dfrac{1}{p^aq^br^c}\), find \(a\), \(b\) and \(c\).

Given that

\(7^x\times49^y=1\)

and

\(5^{5x}\times125^{\frac23y}=\frac1{25},\)

calculate the value of \(x\) and of \(y\).

(a) Solve

\(3^{\left(\frac{x}{2}-1\right)}=10.\)

(b) Solve

\(2e^{1-2y}=3e^{3y+2}.\)

Solve

(i) \(2^{3x-1}=6,\)

(ii) \(\log_3(y+14)=1+\frac{2}{\log_y3}.\)

Solve the simultaneous equations

\(\frac{8^{p+1}}{4^q}=2^{11}, \qquad \frac{3^{2p+5}}{27^{1/3}}=9^{3q}.\)

(a) Given that \(a^7=b\), where \(a\) and \(b\) are positive constants, find

(i) \(\log_a b\),

(ii) \(\log_b a\).

(b) Solve the equation \(\log_{81} y=-\dfrac14\).

(c) Solve the equation

\(\frac{32^{x^2-1}}{4^{x^2}}=16.\)

(a) Solve the equation \(7^{2x+5}=2.5\), giving your answer correct to \(2\) decimal places.

(b) Express

\(\frac{(5\sqrt q)^3}{(625p^{12}q)^{1/4}}\)

in the form \(5^ap^bq^c\), where \(a\), \(b\) and \(c\) are constants.